Why are harmonic functions called harmonic functions? Are they related to harmonic series in any way? Or something else? Wikipedia didn't help.
 A: Consider a sheet of skin stretched into a flat drum head and drummed upon. When the drum head is in vibration, let $f(x,y,t)$ be the height of the drum head at position $(x,y)$ and time $t$. Then $f$ obeys the wave equation:
$$\frac{\partial^2}{\partial t^2} f = c^2 \left( \frac{\partial^2}{\partial x^2} f + \frac{\partial^2}{\partial y^2} f \right) \quad (\ast) $$
where $c$ is a physical constant related to things like how tight the skin is stretched and what it is made out of. Such a solution must also obey the physical constraint that there is no motion at the boundary of the drum, where the skin is nailed down.
Every sound can be composed into its overtones. A pure overtone with frequency $\omega$ corresponds to a solution to the wave equation which looks like $f(x,y,t) = g(x,y) \cos(\omega t+b)$ where 
$$- \frac{\omega^2}{c^2} g =\frac{\partial^2}{\partial x^2} g + \frac{\partial^2}{\partial y^2} g \quad (\ast \ast).$$
Therefore, to understand the sound of a drum, one should figure out for which $\omega$ the PDE $(\ast \ast)$ has solutions which are zero on the boundary of the drum. This is called computing the spectrum of the drum, and a property of the drum which depends only on these $\omega$'s is called a property which one "can hear".
The lowest frequency, which will give the fundamental tone of the drum, will correspond to the lowest nonzero $\omega$ for which $(\ast \ast)$ has solutions. Of course, $(\ast \ast)$ always has the solution that $g$ is a constant and $\omega =0$.
OK, so far that made sense. Now the terminology does something illogical. The name "harmonic" is attached not to the lowest nontrivial frequency, but to the zero frequency. That is to say, $g$ is called "harmonic" if it obeys
$$0=\frac{\partial^2}{\partial x^2} g + \frac{\partial^2}{\partial y^2} g \quad (\ast \ast \ast).$$
I don't know the actual history here, but I think of this as a form of mathematical obtuseness. "You musicians want to study the lowest frequency of vibration? Well you can't get lower than zero!"
The actual physical question addressed by $(\ast \ast \ast)$ is "what are the possible stable shapes for a drumhead, if the boundary is not planar? So, if the rim of my drum varies in height, but I tack the drumhead to it anyway, what shape will the drumhead sit at when we're not pounding on it? This is the Dirichlet problem for the Laplace equation; if I give you the values of a harmonic function on the boundary, what does the interior look like? 
A: I think the connection is that both are connected to the study of the vibrations of a taut string, with precedents dating back to ancient Greece. The Greeks discovered that plesant-sounding (harmonic!) tone intervals were related to small-integer ratios between the dimensions of the producers of the sound.
For an idealized string instrument, the wavelength of the fundamental modes of vibrations are $\frac{2L}{n}$, $n\in \mathbb N^+$. These are also the terms of a harmonic series, hence the name of the latter.
In another direction, the study of oscillations of more general physical objects than strings leads to partial differential equations that are variations of the wave equation. Harmonic functions then encode the relative amplitudes at different places in the object for each mode of vibration.
A: While the term "harmonic" applied to a sequence is ancient, the other usages are more modern.
W. Thompson seems to have been responsible for "spherical harmonic" or "spherical harmonic function", or "spherical surface-harmonic function".  These appear to pre-date the more general "harmonic function"  I find references as early as:
"Philosophical Transactions Of The Royal Society Of London", v. 153, (1863)
https://archive.org/details/in.ernet.dli.2015.26945/page/n659/mode/2up
But there are reports he was using the term a decade earlier.  The latter term was at that time an innovation, however, because he says that formerly they were known in English literature as "Laplace's functions".
The term "harmonic function" was (it seems) coined by H. Poincaré (Actually, he called them "fonctions harmoniques").  See pp. 87-88 of
"Sur les équations de la physique mathématique" 
*Rendiconti del Circolo Matematico di Palermo* (1952 -) 
1894 / 12 Vol. 8; Iss. 1  pp 57-155 
doi 10.1007/bf03012493

He seems to claim that he invented the term.  He does not expound there on why he chose it.  But the reasoning is obvious.  More-or-less as the other responders suggested:

*

*The harmonic sequence is so named because it is exactly the sequence of points on a taut string that deliver musical "harmonics" when the string is touched there as it is plucked.


*The 1-dimensional eigenfunctions of the operator $\Delta$, that is, solutions of the equation
$$
\Delta f = -k^2f
$$
are just the functions describing the motion of an idealized taut string, with the condition that the string is held immobile at the two ends.


*The 2-dimensional flat analog is formally the same equation, but it has a greater variety of boundary conditions, corresponding to differently-shaped drum heads.  (You can, in fact, hear those harmonics.)
Any solution of the equation in any number of dimensions is therefore called "harmonic".


*There are also "spherical harmonics", which satisfy a similar equation, and correspond to the motion of an elastic sphere or a thin spherical shell.


*The term "harmonic analysis" -- I haven't researched yet.  But it is a rather different thing, whereby a function is analyzed into an orthogonal basis set of sinusoidal functions.  (In the special case solutions of the above boundary value problems, the basis might be exactly harmonic functions.)  The meaning of this term has been extended to any analysis into functions or distributions onto an orthogonal basis... so the connection to the original term is only historical.
Overall, the answer to your question is: yes, they are related, and they are related conceptually to the musical term.
